

A141821


Least number k < n and coprime to n such that the largest term of the continued fraction of k/n is as small as possible.


5



1, 2, 3, 2, 5, 5, 3, 7, 3, 8, 5, 5, 11, 4, 7, 12, 13, 7, 9, 8, 17, 7, 7, 7, 19, 19, 23, 12, 11, 12, 25, 10, 13, 27, 11, 10, 9, 14, 11, 29, 11, 31, 31, 19, 17, 34, 37, 18, 19, 40, 41, 14, 17, 21, 15, 16, 17, 18, 47, 17, 23, 46, 45, 46, 25, 49, 49, 50, 29, 26, 19, 27, 31, 29, 55, 34, 61
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OFFSET

2,2


COMMENTS

See A141822 for the value of the largest term in the continued fraction of a(n)/n. Zaremba conjectured that the largest value is 5.


REFERENCES

R. K. Guy, Unsolved problems in number theory, F20.
S. K. Zaremba, ed., "Applications of number theory to numerical analysis," Proceedings of the Symposium at the Centre for Research in Mathematics, University of Montreal, Academic Press, New York, London (1972).


LINKS



EXAMPLE

For n=7, the six continued fractions for k/7 are (0, 7), (0, 3, 2), (0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) and (0, 1, 6). It is easy to see that the fifth one, for 5/7, has the smallest maximum term, 2. Hence a(7)=5.


MATHEMATICA

Table[k=Select[Range[n1], GCD[ #, n]==1&]; c=ContinuedFraction[k/n]; mx=Max/@c; mn=Min[mx]; k[[Position[mx, mn, 1, 1][[1, 1]]]], {n, 2, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



